Continuous-variable quantum key distribution with non-Gaussian quantum catalysis

The non-Gaussian operation can be used not only to enhance and distill the entanglement between Gaussian entangled states, but also to improve the performance of quantum communications. In this paper, we propose a non-Gaussian continuous-variable quantum key distribution (CVQKD) by using quantum catalysis (QC), which is an intriguing non-Gaussian operation in essence that can be implemented with current technologies. We perform quantum catalysis on both ends of the Einstein-Podolsky-Rosen pair prepared by a sender, Alice, and find that for the single-photon QC-CVQKD, the bilateral symmetrical quantum catalysis performs better than the single-side quantum catalysis. Attributing to characteristics of an integral within an ordered product of operators, we find that the quantum-catalysis operation can improve the entanglement property of Gaussian entangled states by enhancing the success probability of non-Gaussian operation, leading to the improvement of the QC-CVQKD system. As a comparison, the QC-CVQKD system involving zero-photon and single-photon quantum catalysis outperforms the previous non-Gaussian CVQKD scheme via photon subtraction in terms of a secret key rate, maximal transmission distance, and tolerable excess noise.

Figure 1

Schematic of the non-Gaussian operations. (a) The QC. An $n$-photon Fock state $|n\rangle$ in auxiliary mode C is split on the asymmetrical BS with transmittance $T_2$. Subsequently, $n$ photons at the auxiliary mode are registered by an ideal photon number resolving detector (PNRD), which is the so-called $n$-photon quantum catalysis represented by an equivalent operator ${\widehat{O}}_n$. (b) The $n$-photon-subtraction operation. Vacuum state $|0\rangle$ in auxiliary mode C is injected into the asymmetrical BS with transmittance $T_2$. Likewise, $n$ photons at the auxiliary mode are detected by the ideal PNRD.

Figure 2

Schematic of QC-CVQKD. Einstein-Podolsky-Rosen (EPR): two-mode squeezed vacuum state. $|m\rangle$ and $|n\rangle$: $m$-photon and $n$-photon Fock states. ${\mathrm{PD}}_{\mathrm{I}}$ and ${\mathrm{PD}}_{\mathrm{II}}$: photon detector by conditional measurement of $|m\rangle$ and $|n\rangle$. $B\left(T_1\right)$ and $B\left(T_2\right)$: ${\mathrm{BS}}_{\mathrm{I}}$ operator with transmittance $T_1$ and the ${\mathrm{BS}}_{\mathrm{II}}$ operator with transmittance $T_2.T_c$ and $\varepsilon$: quantum channel parameters.

Figure 3

The success probability $P_d$ of quantum catalysis as a function of $\alpha$ for BSQC ($T_1=T_2=T$ and $m=n\in \left\{0\text{–}2\right\}$) (dashed-dotted line) and SSQC ($T_1=1,T_2=T$, and $n\in \left\{0\text{–}2\right\}$) (dotted line) for $T=0.95$.

Figure 4

The logarithmic negativity of $E_N{\left(\right|\psi \rangle }_{A_1{B}_1})$ as a function of $\alpha$ for BSQC ($T_1=T_2=T$ and $m=n\in \left\{0\text{–}2\right\}$) (dashed-dotted line) and SSQC ($T_1=1,T_2=T$, and $n\in \{0,1,2\}$) (dotted line) for $T=0.95$.

Figure 5

The optimal logarithmic negativity of $E_N{\left(\right|\psi \rangle }_{A_1{B}_1})$ as a function of $\alpha$ for BSQC ($T_1=T_2=T$ and $m=n\in \{0,1\}$) (dashed-dotted line) and SSQC ($T_1=1,T_2=T$, and $n\in \{0,1\}$) (dotted line) for the optimal choice of $T$.

Figure 6

(a) The success probability of implementing between QC and single-photon subtraction in the $(T,\alpha )$ space with different photon-catalyzed numbers. (b) The logarithmic negativity for the resulted state ${|\psi \rangle }_{A_1{B}_1}$ using QC and the single-photon-subtraction-state ${|\mathrm{\Psi }\rangle }_{A{B}_1}$ as well as the TMSV state ${|\mathrm{TMSV}\rangle }_{AB}$ in the $(T,\alpha )$ space with different photon-catalyzed numbers. In (a) and (b), the magenta surface stands for the single-photon-subtraction case. Other surfaces denote $m=n=0$ (blue surface), $m=n=1$ (green surface), $n=0$ (red surface), and $n=1$ (yellow surface).

Figure 7

The asymptotic secret key rates of the QC-CVQKD system (dashed-dotted lines and dotted lines) as a function of the transmission distance for BSQC cases ($T_1=T_2=T$ and $m=n\in \left\{0\text{–}2\right\}$) (dashed-dotted line) and SSQC cases ($T_1=1,T_2=T$, and $n\in \left\{0\text{–}2\right\}$) (dotted line) with $T=0.95$. The variance of the EPR state is $V=20$, the excess noise is $\varepsilon =0.01$, and the reconciliation efficiency is $\beta =0.95$.

Figure 8

(a) The secret key rates of the QC-CVQKD system (dashed-dotted lines and dotted lines) as a function of the transmission distance for the BSQC cases ($T_1=T_2=T$ and $m=n\in \{0,1\}$) (dashed-dotted line) and the SSQC cases ($T_1=1,T_2=T$, and $n\in \{0,1\}$) (dotted line) for the optimal choice of $T$. (b) The secret key rates of the QC-CVQKD system with the optimal $T$. The variance of the EPR state is $V=20$, the excess noise is $\varepsilon =0.01$, and the reconciliation efficiency is $\beta =0.95$.

Figure 9

The maximal tolerable excess noise of the QC-CVQKD system (dashed-dotted lines and dotted lines) as a function of the transmission distance for BSQC cases ($T_1=T_2=T$ and $m=n\in \{0,1\}$) (dashed-dotted line) and SSQC cases ($T_1=1,T_2=T$, and $n\in \{0,1\}$) (dotted line) for the optimal choice of $T$. The variance of the EPR state is $V=20$, and the reconciliation efficiency is $\beta =0.95$.

Figure 10

(a) The maximal secret key rate for $\epsilon =0.01$ and (b) the maximal tolerable excess noise of the QC-CVQKD system (dashed-dotted lines and dotted lines) and the CVQKD with the single-photon subtraction (magenta solid line) as a function of the transmission distance for the BSQC cases ($T_1=T_2=T$ and $m=n=0,1$) (dashed-dotted line) and SSQC cases ($T_1=1,T_2=T$, and $n=0,1)$ (dotted line) for the optimal choice of $T$. The variance of the EPR state is $V=20$, and the reconciliation efficiency is $\beta =0.95$.

Figure 11

The schematic of the Gaussian modulation CVQKD scheme with single-photon subtraction.